The main diagonals AD, BE and CF of a (convex) cyclic hexagon ABCDEF are concurrent, if and only if, the two products of the alternate sides are equal: AB * CD * EF = BC * DE * FA.
BMO dual corollary generalChallenge: Can you explain why (prove that) the result is true?
i) The result can be proved using ratios of similar triangles. A proof of the result and its converse is available on p. 99 of a useful book by Gardner, A.D. & Bradley, C.J. (2005). Plane Euclidean Geometry: Theory and Problems. University of Leeds, The United Kingdom Mathematics Trust (UKMT).
ii) The result is also mentioned & proved in a paper by Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40, and further generalized in Anghel, N. (2016). Concurrency and collinearity in hexagons. 20. 159-171.
Created by Michael de Villiers, 30 October 2010; updated to WebSketchpad, 7 May 2021.